Question

Find the sum of the infinite series whose sequence of partial sums, Sn, is S sub n equals= 10-1/(n+1)

a) 10
b) 0
c) 9.99
d) Sum does not exist

Answer 1

form the first few terms as:
(10 - 1/2) + (10 - 1/3) + (10 - 1/4) + .....
= 19/2 + 29/3 + 39/4 + .....

I hope you can see that Sal Khan's video fits nicely into your problem
and that you find it quite interesting.
https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-6/v/harmonic-series-divergent

btw, I did a computer simulation and found the following:
Sum(100) = 19/2 +29/3 + 39/4 + ... + 109/101 = 4.197279
sum(500) = .... = 5.79482
sum(2000) = ... = 7.178871
sum(5000) = ... = 8.094717
sum(50,000) = 10.39695 , mmmhhhh?
so , what do you think?

Answer 2

actually, if the sequence of partial sums is Sn = 10 - 1/(n+1)
Sn -> 10 as n->∞

Note that we are not talking about the terms Tn of the sequence itself.